Abelian-extensible automorphism-invariant subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Abelian-extensible automorphism-invariant subgroup, all facts related to Abelian-extensible automorphism-invariant subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Suppose $G$ is an abelian group and $H$ is a subgroup of $G$. We say that $H$ is an abelian-extensible automorphism-invariant subgroup of $G$ if, for every abelian-extensible automorphism $\sigma$ of $G$, we have $\sigma (H)=H$.

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression	$H$ is a fully invariant subgroup of $G$ if ...	This means that full invariance is ...	Additional comments
abelian-extensible automorphism $\to$ function	every abelian-extensible automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for abelian-extensible automorphisms
abelian-extensible automorphism $\to$ endomorphism	every abelian-extensible automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a property stronger than the property of being an endomorphism
abelian-extensible automorphism $\to$ automorphism	every abelian-extensible automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for abelian-extensible automorphisms; i.e., it is the invariance property for abelian-extensible automorphism, which is a group-closed property of automorphisms

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Characteristic subgroup of abelian group				|FULL LIST, MORE INFO
Abelian-potentially characteristic subgroup	a characteristic subgroup in some bigger abelian group	abelian-potentially characteristic implies abelian-extensible automorphism-invariant		|FULL LIST, MORE INFO
Subgroup of finite abelian group	subgroup of finite abelian group	(via abelian-potentially characteristic; see finite abelian NPC theorem)		|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Subgroup of abelian group			subgroup of abelian group not implies abelian-extensible automorphism-invariant